My grandfather was born in the United States and was always a citizen. He didn't naturalize, never had a green card, and was 23 years old when his father naturalized. Oh, by the way, that was a few months after my grandfather's brother was killed in Normandy. I guess to you he wasn't a citizen either when he died for this country.
Naturalization Oath
Maybe you can point to the part of your link that says your adult children are also naturalized by this oath.
How was your great-grandfather in the U.S. for more than 23 years before becoming a U.S. citizen?
And if your mother never applied for her Italian citizenship, she wasn't an Italian citizen - not to mention the fact that she probably never met the additional requirement of serving in the Italian armed forces, becoming an employee of the Italian Government, or living in Italy for two years after reaching legal age. There would have been no Italian citizenship for her to maintain.
Most Italian aliens who served in WWII were naturalized by an act of Congress. If gramps was in the Army, Happy Birthday Mr President Ho-Tep.